Chapter 03 scatterplots and correlation
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This chapter discusses exploring relationships between two quantitative variables using scatterplots and measuring the strength of linear relationships using correlation coefficients. Scatterplots show the joint distribution of two variables by plotting one on the x-axis and the other on the y-axis. Interpreting scatterplots involves examining the overall pattern, direction, strength, and outliers of the relationship. Correlation coefficients measure the direction and strength of linear relationships between -1 and 1. The chapter also discusses adding categorical variables to scatterplots and facts about correlation such as its sensitivity to outliers.
Chapter 03 scatterplots and correlation
- 1. STAT 3615: BIOLOGICAL STATISTICS Hamdy F. F. Mahmoud, PhD Collegiate Assistant Professor Statistics Department @ VT Chapter 3: Scatterplots and correlation
- 2. PART I: EXPLORING DATA VARIABLES AND DISTRIBUTIONS Chapter 1: Picturing distributions with graphs Chapter 2: Describing distributions with numbers RELATIONSHPS Chapter 3: Scatterplots and correlation Chapter 4: Regression Chapter 5: Two-way tables
- 3. CHAPTER 3 TOPICS 3 Bivariate data – response and explanatory variable Scatterplots – two quantitative variable Interpreting the Scatterplots Adding categorical variables to scatterplots – three variables, two quantitative and one categorical. Measuring the association by correlation coefficient, r. Facts about correlation In chapter 1 and 2, we worked with only one variable and summarized it graphically and numerically. In this chapter, we cover relationship between two quantitative variables and describe it graphically and measure the association numerically
- 4. For each individual studied, we record data on two variables. We then examine whether there is a relationship between these two variables. ❖ Do changes in one variable tend to be associated with specific changes in the other variables? Bivariate Data – response and explanatory variable In this table at the left, we have two quantitative variables recorded for each of 12 students: • How many beers they drank. • Their resulting blood alcohol content (BAC). Number of Beers Blood Alcohol Content 2 0.03 7 0.09 3 0.07 4 0.07 5 0.08 8 0.12 3 0.04 5 0.06 6 0.10 7 0.09 1 0.01 4 0.05
- 5. – A response variable measures an outcome of a study. An explanatory variable may explain or influence changes in a response variable. – Examples: • Number of beer and blood alcohol content • Age and an animal weight • Corn yield and amount of rain. Explanatory variable Response variable May affect Bivariate Data – response and explanatory variables
- 6. In each of the following situations, is it more reasonable to simply explore the association between the two variables or to view one of the variables as an explanatory variable and the other as a response variable? a) The typical amount of calories a person consumes per day and that person’s percentage of body fat. b) The weight in kilograms and height in centimeters of a person. c) Inches of rain in the grown season and the yield of corn in bushels per acre. d) A person’s leg length and arm length, in centimeters. Practice on response and explanatory variables
- 7. •A scatterplot shows the relationship between two quantitative variables measured on the same individuals. •The values of one variable appear on horizontal axis (explanatory variable) and the values of the other variable appear on the the vertical axis (response variable). Scatterplots – two quantitative variables
- 8. Example: An endangered species Manatees are large, herbivorous, aquatic, endangered mammals found primary in the rivers and estuaries of Florida. Research question: • Do you think powerboats are responsible to manatees death?
- 9. Powerboat registrations (in thousands) and manatee deaths from powerboat collisions in Florida Example: An endangered species (cont.) A study examined the relationship between the number of manatee deaths from powerboat collisions and the number of powerboats registered in any given year between 1977 and 2012.
- 10. Scatterplot of the number of manatee deaths due to powerboat collisions in Florida each year against the number of powerboats registered (in thousands) that same year. The dotted lines intersect at the point (755, 54), the data for year 1997. • Do you think powerboats are responsible to manatees death?
- 11. Do you think there is an association between Brain size and brain performance measured by IQ ? 1100000 1050000 1000000 950000 900000 850000 800000 160 150 140 130 120 110 100 90 80 70 Brain Size (pixels) Performace IQ Scatterplot of Performace IQ vs Brain Size (pixels) Comments:
- 12. To interpret a scatterplot, you need to look at: Overall pattern (Form): linear or nonlinear Direction: if it is linear, positive or negative Strength of the relationship which is determined by how close the points in the scatterplot making a specific pattern or form. Outliers: individuals values that fall outside the overall pattern of the relationship. Interpreting the Scatterplots
- 13. The pattern (form) of the relationship between 2 quantitative variables refers to the overall pattern. 100 90 80 70 60 50 140 130 120 110 100 90 80 70 60 Temperature Mortality Scatterplot of Mortality vs Temperature
- 14. Positive association: High values of one variable tend to occur together with high values of the other variable. Negative association: High values of one variable tend to occur together with low values of the other variable. • If the relationship is linear, look at the direction
- 15. The strength of the relationship between 2 quantitative variables refers to how much variation, or scatter, there is around the main form.
- 16. An outlier is a data value that has a very low probability of occurrence (i.e., it is unusual or unexpected). In a scatterplot, outliers are points that fall outside of the overall pattern of the relationship.
- 17. In many cases, the relationship is not between the two variables, but it is between the transformed variables. In other words, the relation is not between x and y, but it may be between x and square root of y, log(x) and log(y), …. etc. Transformed variables and association
- 18. Is there a relationship between brain size and body weight among animals? To answer this research question, a researcher collected data for 62 different types of animals. Excel file below shows a part of this data. Example: body weight and brain size
- 19. Is there a relationship between brain size and body weight among animals? 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000 7000 Brain weight Body weight Brain weight and body weigt scatter plot It seems there are some outliers or unusual values! Example: body weight and brain size (cont.)
- 20. Is there a relationship between brain size and body weight among animals? Example: body weight and brain size (cont.) 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 Brain weight Body weight Brain weight and body weight After removing the outliers.
- 21. Is there a relationship between brain size and body weight among animals? Example: body weight and brain size (cont.) After taking logarithm of both variables. -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 -3 -2 -1 0 1 2 3 log Brain weight log Body weight Brain weight and body weight
- 22. Adding a categorical variable to a scatterplot ➢Two or more relationships can be compared on a single scatterplot when we use different symbols for groups of points on the graph. The graph compares the association between thorax length and longevity of male fruit flies that are allowed to reproduce (green) or not (purple). The pattern is similar in both groups (linear, positive association), but male fruit flies not allowed to reproduce tend to live longer than reproducing male fruit flies of the same size.
- 23. • Energy expended as a function of running speed for various treadmill inclines However, for each incline, there is a very strong, positive, linear relationship between energy expenditure and speed. In addition, we find that the relationship between energy expenditure and speed is noticeably different for different inclines: More energy tends to be expended for a given running speed if the incline is steeper (uphill).
- 24. The correlation measures the direction and strength of the linear relationship between two quantitative variables. Suppose we have data on variables x and y for n individuals. The correlation r between x and y is Or you can use Where is the average or x is the average of y is the standard deviation of x is the standard deviation of y x y sx sy Measuring the Association by Coefficient of Correlation, r
- 25. • r ranges from −1 to +1 • Strength is indicated by the absolute value of r • Direction is indicated by the sign of r (+ or –) -1£ r £1
- 26. Blood alcohol content (BAC) a) Which is explanatory and which is response? b) Draw a scatterplot and interpret. c) Calculate the coefficient of correlation, r. To examine the association between number of beers the person drank and the resulting blood alcohol content (BAC), 16 students have been surveyed and the data are recorded. Number of Beers Blood Alcohol Content 2 0.03 7 0.09 3 0.07 4 0.07 5 0.08 8 0.12 3 0.04 5 0.06 6 0.10 7 0.09 1 0.01 4 0.05
- 27. 1 2 3 4 5 6 7 8 9 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Number of beers Blood Alcohol Content Number of Beers Blood Alcohol Content 2 0.03 7 0.09 3 0.07 4 0.07 5 0.08 8 0.12 3 0.04 5 0.06 6 0.10 7 0.09 1 0.01 4 0.05
- 28. Calculate the coefficient of correlation, r.
- 30. FACTS ABOUT CORRELATION ❑ Correlation makes no distinction between explanatory and response variable. ❑ Because r uses standardized values of the observations, it does not change when we change units of measurement of x, y, or both. ❑ Positive r indicates positive association and negative r indicates negative association between variables. ❑ The correlation r is always a number between -1 and 1. Values near to 0 indicates a very weak linear relationship. -1£ r £1
- 31. ❑Correlation requires that both variables be quantitative. ❑Correlation measures only linear relations, it does not describe curved relationships. • Correlations are calculated using means and standard deviations, and thus are NOT resistant to outliers. Let us play with scatterplot Click Like mean and standard deviation it is not resistant to outliers.
- 33. End of Chapter 3